resonance

[ElDavidas]

In a question, I've been asked to find the general solution of the equation:

\ddot{x} + \omega^2{}_{0} = cos( \omega t)

(\omega , \omega_{0} > 0, \omega \neq \omega_{0})

I've found the solution to this.

It then asks me if this solution exhibits resonance. What does this mean and how do you determine this?

[Tide]

If you found the solution then it should be apparent whether it exhibits resonance. It might be helpful if you displayed your solution here.

Incidentally, if you have excluded the possibility of \omega = \omega_0 then you have excluded the possibility of [exact] resonance! :)

[ElDavidas]

It might be helpful if you displayed your solution here.


Ok, this is my answer:

x = A sin( \omega_{0}t) + B cos (\omega_{0}t) + \frac {1} {\omega_0^2 - \omega^2} cos (\omega t)

[inha]

The denominator of the last term is what you're after. What happens to the solution as \omega -> \omega_0? I'll try to find you a legendary video of resonant behaviour.


edit: here's a small clip http://www.camerashoptacoma.com/mpegs/TacomaNarrowsBridge.mpg